Let $n, r$ and $f$ be positive integers. Let $p$ be a prime number and $psi$ be an arbitrary fixed nontrivial additive character of the finite field $mathbb F_q$ with $q=p^f$ elements. Let $F$ be a polynomial in $mathbb F_qx_1,dots ,x_n$ and $V$ be the affine algebraic variety defined over $mathbb {F}_q$ by the simultaneous vanishing of the polynomials ${F_i}_{i=1}^rsubseteq mathbb F_qx_1,dots ,x_n$. Let $mathbb {Z}_{ge 0}$ stand for the set of all nonnegative integers and $A$ be an arbitrary nonempty subset of ${1,dots ,n}$. For a polynomial $H(X)=sum _{{mathbf {d}}}alpha _{mathbf {d}}X^{mathbf {d}}$ with ${mathbf {d}}=(d_1,dots ,d_n)in mathbb {Z}_{ge 0}^n, X^{mathbf {d}}=x_1^{d_1}dots x_n^{d_n}$ and $alpha _{mathbf {d}}in mathbb {F}_q^*$, we define $deg _A(H)=max _{{mathbf {d}}}{sum _{iin A}d_i}$ to be the $A$-degree of $H$. In this paper, for the exponential sum $S(F,V,psi )=sum _{Xin V(mathbb {F}_q)}psi (F(X))$ with $V(mathbb {F}_q)$ being the set of the $mathbb {F}_q$-rational points of $V$, we show that begin{equation*} mathrm {ord}_q S(F,V,psi )ge frac {A-sum _{i=1}^rdeg _A(F_i)} {max _{1le ile r}{deg _A(F),deg _A(F_i)}} end{equation*} if $deg _A(F)>0$ or $deg _A(F_i)>0$ for some $iin {1,dots ,r}$. This estimate improves Sperber’s theorem obtained in 1986. This also leads to an improvement of the $p$-adic valuation of the number $N(V)$ of $mathbb {F}_q$-rational points on the variety $V$ which strengthens the Ax-Katz theorem. Moreover, we use the $A$-degree and $p$-weight $A$-degree to establish $p$-adic estimates on multiplicative character sums and twisted exponential sums which improve Wan’s results gotten in 1995.
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